The fundamental idea was to add further axioms to the system, in such a way that the 'true but unprovable' statements could be proved.
基本的想法就是在系统中加入更多的公理，使不可证明的命题变成可证明的。
It was easy enough to add an axiom so that one of Gdel's peculiar statements could be proved.
添加一条公理是很容易的，可以使一条命题变得可证明，
But then Gdel's theorem could be applied to the enlarged set of axioms, producing yet another 'true but unprovable' assertion.
但问题是，哥德尔定理同样适用于扩大后的公理集，所以又会产生新的不可证明的命题。
It could not be enough to add a finite number of axioms; it was necessary to discuss adding infinitely many.
因此，加入有限多个公理是不够的，必须要讨论另一种情形，那就是加入无限多个公理。
This was just the beginning, for as mathematicians well knew, there were many possible ways of doing 'infinitely many' things in order.
这仅仅是个开始，数学家们都知道，有很多方法可以处理无限问题。
Cantor had seen this when investigating the notion of ordering the integers.
康托在研究整数的次序时，就考虑到了这一点。
Suppose, for example, that the integers were ordered in the following way:
他假设说，如果把整数这样排列：
first all the even numbers, in ascending order, and then all the odd numbers.
首先是所有的偶数，按升序排列，然后是所有的奇数。
In a precise sense, this listing of the integers would be 'twice as long' as the usual one.
从直觉上来说，这样排成的序列，应该是正常顺序的两倍长，
It could be made three times as long, or indeed infinitely many times as long, by taking first the even numbers, then remaining multiples of 3, then remaining multiples of 5, then remaining multiples of 7, and so on.
同理还可以排成三倍长，甚至可以是无限倍长。
Indeed, there was no limit to the 'length' of such lists.
总之，这个序列的长度是无限的。